Melt butter in a large pot over medium heat. Add onion, carrot, and garlic, and sauté until soft.
Sprinkle in flour, stirring constantly for about 2 minutes to form a roux.
Slowly whisk in the chicken broth and beer, ensuring a smooth consistency.
Bring to a simmer and cook for about 10 minutes, stirring occasionally.
Reduce heat to low and gradually add shredded cheddar cheese, stirring until fully melted.
Stir in heavy cream, Dijon mustard, Worcestershire sauce, salt, cayenne pepper, and black pepper.
Simmer for an additional 5 minutes, then remove from heat.
Serve hot, garnished with croutons or crispy bacon bits.